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Math Help - Polynomial Generation

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    Polynomial Generation

    Construct a polynomial of the fourth degree whose coefficients are rational numbers,one of whose roots is 2i [i = square root of (-1)] and which has two irrational roots.
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by aravindv01 View Post
    Construct a polynomial of the fourth degree whose coefficients are rational numbers,one of whose roots is 2i [i = square root of (-1)] and which has two irrational roots.
    In a rational polynomial, complex roots always come in pairs of conjugates (because of the formula \frac{-b \pm \sqrt{b^2-4ac}}{2a}).

    So, as 2i is a root its complex conjugate is also a root. What is the complex conjugate of 2i?

    Now, there is a rational polynomial of the form ax^2+bx+c with these two complex numbers as roots. What is it?* This will then give you your answer...

    *A hint for this bit would be that if p and q are roots of a quadratic polynomial P then P=(x-p)(x-q). Then expand.
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